A problem play is one that deals with contentious social issues, often through debates between the various characters. The Merchant of Venice could be said to fall into this category, although problem plays as we know them today only really emerged in the late 19th century, in the work of such dramatists as Shaw and Ibsen.
The relevant social problem in Shakespeare's play is the treatment of Jews. The city of Venice is depicted as a place where Jews are routinely subjected to official discrimination. Not only that, but as Shylock tells us in his famous monologue, they are also the regular victims of personal slights and insults from Christians who think they have the right to treat Jews however they please.
But Shylock's resolution of the problem of discrimination is rather unusual, to say the least. He demands that as Jews and Christians are basically equal—they both have eyes, five senses, get sick with the same diseases, bleed if they're pricked, etc—he's every bit as entitled to gain his revenge on a Christian as a Christian would be on a Jew.
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